72 Prof. Burnside, On a configuration of twenty-seven 



there results a configuration of points and hyper-planes ; a certain 

 number of hyper-planes passing through each point and a certain 

 number of points lying in each hyper-plane ; with the restriction 

 that the whole of the points lie on a four-dimensional quadric. As 

 in the previous case this restriction is not a necessary condition 

 for the existence of the configuration. The object of the present 

 note is partly to prove this result in a particularly interesting case, 

 and partly to bring out the tactical analogy (which the numbers 

 suggest) of the configuration of 27 hyper-planes with the con- 

 figuration of 27 lines on a cubic surface. 



To avoid the continual use of the word hyper-plane, a flat 

 manifold in four-dimensional space, determined by four points, is 

 called simply a plane ; and when it is necessary to refer to one 

 determined by three points it is called a three-dimensional plane. 



A quadric in four-dimensional space is determined by fifteen 

 points. Eleven points therefore determine four linearly inde- 

 pendent quadrics. Every one of a system of quadrics through 

 eleven points must therefore have five other common points. 

 This and the fact that four points determine a plane and four 

 planes determine a point forms the basis of the reasoning. 



1. In a four-dimensional space, consider a base-point 0, and 

 five planes, 



1, 2, 3, 4, 5, 



passing through it. On the line of intersection of each set of 

 three planes, mark a point distinct from ; and denote the 

 point on the line of intersection of 1, 2 and 3 by 123. (In this 

 symbol the sequence of the figures is immaterial.) Denote the 

 plane which passes through 123, 124, 134, and 234 by 1234. 

 (In this symbol again the sequence of the figures is immaterial.) 

 Then the five pairs of planes 



1,2345; 2,1345; 3,1245; 4,1235; 5,1234, 



are a set of quadrics through 11 points, viz. and the 10 points 

 123, etc. Hence they determine a set of 5 further points, which 

 lie on all the quadrics, so that 8 of the 16 lie in each plane. Of 

 these 5 then, one must lie on 1 and the other four on 2345. That 

 one of the five which lies on 1 cannot also lie on 2 ; and a suitable 

 notation for the five points in what follows will be 



16, 26, 36, 46, 56, 



(here the sequence of the figures is essential), where 16 lies in the 

 five planes 



1, 1345, 1245, 1235, 1234. 



The figure thus constructed is a complete one in the sense that 

 each point (and each plane) bears the same relation to the figure 



